Theoretical model: Achievable pulse width for two special cases

The temporal width of pulses generated by temporal compression of the sweeps of wavelength-swept light sources inevitably depends on the coherence properties of the source. In order to be able to judge the results obtainable with an FDML laser, it is rea- sonable to consider the two special cases of fully incoherent5 and fully coherent wave- length-swept light sources.

4.3.1.1 Fully incoherent case

In the following, for the sake of simplicity, we assume a fully incoherent wavelength- swept light source that sweeps with a constant sweep speed . The light source is expected to have a constant instantaneous linewidth . The minimum achievable pulse width after optimal temporal compression then is determined by two effects:

On the one hand, the pulse width is defined due to the time-bandwidth limitation regard- ing the instantaneous linewidth . The latter directly determines the instantaneous coherence time which corresponds to a minimum potential pulse width , defined as:

5 _{Note that, in the context of this thesis, the terminology “fully incoherent” is used for wavelength-swept}

light sources that have no optical feedback (narrowband filtering of spectrally broadband incoherent light, no lasing).

. Here, a Gaussian spectral shape of the instantaneous spec- trum is assumed and c is the speed of light in vacuum. The reason is as follows: Since the phase relation of the electric field of the spectral contributions within each sweep and between different sweeps is arbitrary, an optimal temporal compression equally applied to each forward sweep increases the instantaneous power but cannot yield a reduction of the pulse widths below . In Figure 4.4a, b and c, is plotted against the instantaneous linewidth (black dashed curve).

On the other hand, the minimum achievable pulse width is influenced by the wave- length sweep speed . Due to wavelength sweeping, the resolvable spectral win- dow of the instantaneous linewidth directly translates to a resolvable temporal window . In the fully incoherent case and assuming

, there is a random phase relation in the electric field evolution within each time interval , as well as between different adjacent time intervals . An optimal temporal compression equally applied to each forward sweep increases the instantane- ous power but cannot yield a reduction of the pulse width below . In Figure 4.4a, b and c, is sketched for two distinct wavelength sweep speeds corresponding to 4x pass of the DCF (cyan dashed line) and 1x pass of the DCF (pink dashed line). Consequently, on the one hand, the minimum achievable pulse width equals if the instantaneous linewidth is very small and/or the sweep speed is very high. On the other hand, equals if the instantaneous linewidth is very large and/or the sweep speed is very small. A more general solution, including both effects, can be obtained by convolution (assuming Gaussian shape). The minimum

Figure 4.4: a. Results of a theoretical model describing temporal compression of the wavelength- swept output of fully incoherent wavelength-swept light sources: Minimum achievable pulse width against the instantaneous linewidth of the swept light source in case of two different filter sweep speeds corresponding to 4x pass of the DCF (blue curve) and 1x pass of the DCF (violet curve). Both curves are defined on the basis of two contributions: due to time-bandwidth limitation (black dashed line) and due wavelength sweeping (4x pass of the DCF: cyan dashed line and 1x pass of the DCF: pink dashed line). The crosses show experimental results obtained with an incoherent wavelength-swept ASE source (4x pass of the DCF) for different sweep bandwidths . b. and c. Zoomed views of a, corresponding to two different filter sweep speeds. The green area (4x pass of the DCF) and the red area (4x pass of the DCF) show the parameter range corresponding to partially coherent superposition of the different spectral components with- in the sweeps. Black ovals indicate the approximate parameter range currently achievable with FDML lasers, clearly included in the area representing partially coherent superposition.

0 10 20 30 40 50 60 200 400 600 1x pass DCF F W H M o f p u ls e [p s] inst. linewidth (FWHM) [pm] 0 100 200 300 1 2 3 4  = 5nm  = 10nm F W H M o f p u ls e [n s] inst. linewidth (FWHM) [pm] 0 10 20 30 40 50 60 200 400 600 4x pass DCF F W H M o f p u ls e [ p s] inst. linewidth (FWHM) [pm] a. b. FDML partially coherent part. coh. FDML c.

achievable pulse width in the fully incoherent case can then be estimated to:

4.1

This limit is illustrated in Figure 4.4a, b and c in case of sweep speeds corresponding to 4x pass of the DCF (blue curve) and 1x pass of the DCF (violet curve). Consequently, one can draw the following conclusion: If the temporal compression of the sweeps from a wavelength-swept laser with a distinct instantaneous linewidth enables a pulse width , there must occur at least partially coherent superposition of the different spectral components of the sweeps within the temporal compression pro- cess. The parameter range of partial coherence is illustrated in Figure 4.4b (green area) in case of 4x pass of the DCF and in Figure 4.4c (red area) in case of 1x pass of the DCF.

4.3.1.2 Fully coherent case

In the fully coherent case, the minimum temporal pulse width is only determined by the sweep bandwidth . The electric field within the each sweep and between adja- cent sweeps is optimally coherent resulting in perfect mode locked laser operation. In this case, all longitudinal laser modes exhibit a defined phase relation. Assuming a Gaussian spectral shape, i.e. a time-bandwidth product of 0.44, the minimum pulse width reads:

4.2

With regard to the sweep bandwidths 6 nm (1x pass of the DCF) and 1.5 nm (4x pass of the DCF), as chosen for the experiments within the framework of this thesis, the corresponding minimum pulse widths in the fully coherent case are 600 fs and

2.4 ps, respectively (assuming a time-bandwidth product of 0.44).